[1]}
Let \(f=a/b\in F(z)\)  be such that \(a, b\in F[z]\)  and \(\gcd (a, b)=1\) . Then \(f\)  is \( {\sigma }_z\) -summable in \(F(z)\)  if and only if
\({\operatorname{res}}_{ {\sigma }_z}(f, d, j)=0\)  for any irreducible factor \(d\)  of the denominator \(b\)  of any multiplicity \(j\in  {\mathbb {N}}\) .

By Abramov's reduction [2]}, [3]}, we can decompose a rational function \(f\in F(z)\)  as
\(f = \Delta _z(g) + \sum _{i=1}^n \sum _{j=1}^{m_i}\frac{a_{i, j}}{b_i^j},\) 
